Application of One Server Queuing Models to Customers Management in the Cafeteria A Case Study of...

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Application of One ServerManagement in t

Babalola University, AdoS. O. Adejuwon M. Tech

1,3

1,2Department of mathematical and physical sciences, Afe3Department of Statistics Ekiti State University, Ado Ekiti

ABSTRACT This paper seeks to clarify the impact of time wasting on the weak performance, efficiency and effectiveness of their operations at Abuad cafeteria by applying queuing theory in a real life situation. We obtained the data from the major cafeteria in Abuad.derive the arrival rate, service rate, utilization rate, waiting time in queue. Based on the data Little’s Theorem and M/M/1 queuing model was also considered. The arrival rate at at Abuad cafeteria during its busiest period of the day is 2.92 cusper minute (cpm) while the service rate is 2.95 cpm. The average number of customers in the cafeteria is 219 and the utilization period is 0.995. We conclude the paper by given the concluding remark and acknowledge. KEY WORDS: Abuad, cafeteria, waiting lines, Little’s theorem, Utilization factor, customers. 1. INTRODUCTION Queuing theory is the mathematical study of waiting lines which are the most frequently encountered problems in everyday life. For instancecafeteria, telecommunication, library, bank, etc. Common to all of these cases are the arrivals of objects requiring service and the attendant delays when the service mechanism is busy. Waiting lines cannot be eliminated completely, but suitable techniques can be used to reduce the waiting time of an object in the system. A long waiting line may result in loss of customers or loss of goodwill toorganization. Waiting time can be reduced by providing additional service facilities, but it may result in an increase in the idle time of the service

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ne Server Queuing Models to CustomersManagement in the Cafeteria: A Case Study of Afe

Babalola University, Ado-Ekiti (Abuad)

Tech1, V. G. Jemilohun M. Sc2, A. O. Ilesanmi1,3Assistance Lecturer, 2Lecturer

Department of mathematical and physical sciences, Afe Babalola University, AdoDepartment of Statistics Ekiti State University, Ado Ekiti, Nigeria

This paper seeks to clarify the impact of time wasting on the weak performance, efficiency and effectiveness of their operations at Abuad cafeteria by applying queuing theory in a real life situation. We obtained the data from the major cafeteria in Abuad. We then derive the arrival rate, service rate, utilization rate, waiting time in queue. Based on the data Little’s Theorem and M/M/1 queuing model was also considered. The arrival rate at at Abuad cafeteria during its busiest period of the day is 2.92 customers per minute (cpm) while the service rate is 2.95 cpm. The average number of customers in the cafeteria is 219 and the utilization period is 0.995. We conclude the paper by given the concluding remark and

cafeteria, waiting lines, customers.

is the mathematical study of waiting lines which are the most frequently encountered

instance, queue at a library, bank, etc.

Common to all of these cases are the arrivals of objects requiring service and the attendant delays when the service mechanism is busy. Waiting lines cannot be eliminated completely, but suitable

d to reduce the waiting time of an object in the system. A long waiting line may

or loss of goodwill to an organization. Waiting time can be reduced by providing additional service facilities, but it may

the idle time of the service

mechanism. In this paper queuing theory will to the area of cafeteria. A cafeteria is a type of food service location in whichthere is a little or no waiting staff table service, whether a restaurant or within an institution such as a large office building or school.which offers a varied menu and comfortable surroundings enable students to experience a sens‘’home’’ which on campus which can engage in leisurely conversation and interactive activities with their peers (Norhati and NurHafisah, 2013, Raman and Chinniah, 2011).’ There are several factors for a restaurant to be attract customers, the most important are; task, cleanliness, food quality, service quality, restaurant environment and safety are most important factors (2013; Haghighi et’al 2012). These factors, when managed carefully, will be able to customers. However, there is also another factor that needs to be considered especially when the restaurant has already succeeded in attracting customers. This factors is the customers queuing time. Queuing theory is the study of queue or waiting lines. Some of the analysis that can be derived using queuing theory includes the expected queuing length, the probability of the system to be in certain states, such as empty or full (Dharmawraya and Adi, 2011). Queuing theory is very common in respecially during lunch and dinner time. Hence, queuing theory is suitable to be applied in a restaurant setting, since it has an associated queue or waiting

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Oct 2018 Page: 1624

Queuing Models to Customers A Case Study of Afe Ekiti (Abuad)

Ilesanmi M. Tech3

Babalola University, Ado-Ekiti, Nigeria Nigeria

In this paper queuing theory will be apply

A cafeteria is a type of food service location in which there is a little or no waiting staff table service, whether a restaurant or within an institution such as a large office building or school. University cafeteria which offers a varied menu and comfortable surroundings enable students to experience a sense of ‘’home’’ which on campus which can engage in leisurely conversation and interactive activities with

NurHafisah, 2013, Raman

There are several factors for a restaurant to be attract rtant are; task, cleanliness,

food quality, service quality, restaurant environment and safety are most important factors (Mishra et’al

). These factors, when managed carefully, will be able to attract plenty of customers. However, there is also another factor that needs to be considered especially when the restaurant has already succeeded in attracting customers. This factors is the customers queuing time. Queuing theory

waiting lines. Some of the analysis that can be derived using queuing theory includes the expected queuing length, the probability of the system to be in certain states, such as empty or

Adi, 2011).

Queuing theory is very common in restaurants especially during lunch and dinner time. Hence, queuing theory is suitable to be applied in a restaurant setting, since it has an associated queue or waiting

International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456


line where customers who cannot be served immediately have to queue (wait) for service.Researchers have previously used queuing theory to model the restaurant operation, reduce cycle time in a busy fast food restaurant, as well as to increase throughout and efficiency (Brann and Curin et’al 2005; Kharwat, 1991). This paper uses queuing theory to study the waiting lines in ABUAD cafeteria. There arecafeterias in Abuad, these are: cafe 1, café 2 and the staff cafeteria. Cafeteria provides tables of four people. There are 3 to 6 waiters or waitresses workingat any point in time in these entire three daily basis, the three cafes servers over 5000 customers during weekdays, and over 350during weekends. This paper seeks to clarify impact of time wasting on the weak performanceefficiency and effectiveness of their Abuad cafeteria by applying queuing theory in a real life situation. 2. QUEUINGTHEORY Queuing theory is a mathematical approach in operation research applied to the analysis of waiting lines. A.K. Erlang first analyzed queues in 1913contest of telephone facilities. He identified that the number of telephone conversations and telephone holding time put into Poisson distribution and exponentially distributed. The body of knowledge that developed thereafter via research and analysis came to be known as queuing theory, and is extensively applied in industrial settings and retail sectors. 3. LITTLE ’STHEOREM Little ’s (1961) little’s theorem describes the relationship between throughout café (i.e arrival and service rate), cycle time and work in process. The theorem states that the expected numbers of customers L for a system in steady state can be reduce from Little’s theorem.

L = Tλ Whereλ the expected number of customers’ arrival rate and T is the expected service time for a customerThree fundamental relationships can be derived from little ’s theorem (Lag Una & Marklund, 2005)� L increases if λ or T increases � λ increases if L increases or T decreases� T increases if L increases or λ decreases

Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456


cannot be served immediately have to queue (wait) for service. Researchers have previously used queuing theory to model the restaurant operation, reduce cycle time in a busy fast food restaurant, as well as to increase

Kuuck, 2002;

queuing theory to study the waiting lines in ABUAD cafeteria. There are three major

cafe 1, café 2 and the staff cafeteria. Cafeteria provides tables of four

waitresses working three cafes. On a

s servers over 5000 3500 customers

during weekends. This paper seeks to clarify the of time wasting on the weak performance,

d effectiveness of their operations in applying queuing theory in a real

Queuing theory is a mathematical approach in operation research applied to the analysis of waiting

queues in 1913 in the contest of telephone facilities. He identified that the number of telephone conversations and telephone holding time put into Poisson distribution and

The body of knowledge that developed thereafter via earch and analysis came to be known as queuing

theory, and is extensively applied in industrial settings

s theorem describes the hroughout café (i.e arrival and

cycle time and work in process. The theorem states that the expected numbers of customers L for a system in steady state can be reduce

(1)

customers’ arrival ted service time for a customer.

can be derived from s theorem (Lag Una & Marklund, 2005)

f L increases or T decreases decreases

4. ABUAD RESTAURANT QUEUING MODELWe will discuss only the single queue model which is: One server (Queue with M│M│ An M│M│I system is a queuing system exponential distributed inter arrival time, with parameter λ; exponentially distributed service time, with parameter µ, one server, no limits on the system capacity and a queue discipline of first come first served. The constant λ is the average cusrate while the constant µ is the average service rate of customers. Both are in units of customers per time. The expected inter-arrival time and the expected time to serve one customer are 1/λM│M│I systems are often referred to as system with single-server, infinite capacity, queuing systems having Poisson input and exponential service times. We required that:

ρ =ηλ

: utilization factor

Whereλ : the mean customers arrival rate η : The mean service rate Pn: the probability of having n customers Probability of zero customers

Pn= ( ρ−1 ) ρη Po= ρ−1

L: Expected number of customers dining in the restaurant

L=ρ

ρ−1

=λµ

λ−

LV: Expected number of customers

Lv= LxP=ρ

ρ−1

2=

λµρλ−

Expected waiting time of customers in the queues

Wq=λµ

ρ−

Expected waiting time spent in Abuad restinclude the waiting time

Ws=λµ −

1

Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470

Oct 2018 Page: 1625

ABUAD RESTAURANT QUEUING MODEL We will discuss only the single queue model which is:

│M│I model)

I system is a queuing system having exponential distributed inter arrival time, with

; exponentially distributed service time, with parameter µ, one server, no limits on the system capacity and a queue discipline of first come first

is the average customer arrival rate while the constant µ is the average service rate of customers. Both are in units of customers per time.

arrival time and the expected time to serve one customer are 1/λ and 1/µ respectively.

eferred to as system with server, infinite capacity, queuing systems

having Poisson input and exponential service times.

: utilization factor (2)

the mean customers arrival rate

: the probability of having n customers

(3)

(4)

Expected number of customers dining in the

(5)

: Expected number of customers λ the queue

λ (6)

Expected waiting time of customers in the queues

(7)

Expected waiting time spent in Abuad restaurant

(8)



5. RESULTS AND DISCUSSION FOR ABUAD CAFETERIA QUEUING SYSTEM ANALYSIS

The data are obtained from all the three major University restaurants through interview with the restaurant manage as well as data collections through observations at the cafeteria. Some students were also interviewed to know their interest derived from those cafeterias. The daily numbers of visitors were obtained from the cafeteria at managers. The cafeteria has been the data as part of their end of day routine number of tickets sold out to the visitors based on the interview with the cafeteria managers. Below is the one month daily customer record which was collected from the cafeteria

Table 1: one month daily customer Weeks

Mon.

Tues.

Wed.

Thur.

Fri.

1ST Week

2317

2764

3452

3389

4014

2nd Week

3762

3542

4072

4087

4287

3rd Week

4007

3231

3528

4109

4511

4th Week

3276

3974

4121

4275

4713

Total 13362

13511

15173

15860

17525

Fig1. From the fig 1 above it can be deduced that the numbers of customers on Fridays to Sundays are more than the number of customers during the weekdays. The free period for all the students is on weekend so they have time for pleasure.



FOR ABUAD CAFETERIA QUEUING SYSTEM

re obtained from all the three major through interview with the

as well as data collections through . Some students were also

derived from those

The daily numbers of visitors were obtained from the has been collated

end of day routine through the sold out to the visitors based on the

Below is the one month daily customer record which

one month daily customer sale

Fri. Sat. Sun

. 1

4117

4513

8

4592

4762

451

4976

4517

471

4727

4676

17525

18412

18468

fig 1 above it can be deduced that the numbers of customers on Fridays to Sundays are more than the number of customers during the weekdays. The free period for all the students is on weekend so

Fig2. The trend of one month total customers sale is shown the fig.2 above 6. CALCULATION We conducted the research at abuad cafeteria. There are average 4500 people coming to restaurant in 12 hours’ time window of dinner time. We derive arrival rate λ = 4500/ 720

= 6.25customers/min.

We observed & discussed with managers of the cafeteria that each customers spend 35 min. on average in cafeteria (Ws), the queue length Lq is about 47 people on average & waiting time is nearly

15 minutes, using Wq = Wq=

actual waiting time is not very much different when compared to the theoretical waiting time. Wq = 47/6.25 = 7.52 minutes Now, calculating average no. of customers in cafeteria using little’s λWs = 6.25 × 35 min. = 219derive utilization rate & service rate

Ls=ρ

ρ−1

=

L

L)1( += λµ =219

)2191(25.6 +

Hence,ρ =ηλ

=.6

.6

With very high utilization rate of .995 during dinner time, the probability of zero customers in cafeteria is very small as P0=1- ρ = 1-.995=.005 7. CONCLUDING REMARKS � Queuing theory is not new but only recently has

restaurant begun to use it effectively.


Oct 2018 Page: 1626

The trend of one month total customers sale is

e conducted the research at abuad cafeteria. There are average 4500 people coming to restaurant in 12 hours’ time window of dinner time. We derive arrival

customers/min.

We observed & discussed with managers of the cafeteria that each customers spend 35 min. on average in cafeteria (Ws), the queue length Lq is

people on average & waiting time is nearly

Wq=λµ

ρ−

the observed

actual waiting time is not very much different when compared to the theoretical waiting time. Wq =

minutes Now, calculating average no. using little’s formula Ls =

219 customers Now, we derive utilization rate & service rate

λµλ−

=6.28 customers/minute

28.

25. =0.995

With very high utilization rate of .995 during dinner time, the probability of zero customers in cafeteria is

.995=.005

CONCLUDING REMARKS Queuing theory is not new but only recently has restaurant begun to use it effectively.



� The utilization rate at the cafeteria(0.995). This is only utilization rate during lunch and dinner from Monday to Sunday.

� The utilization is directly proportional with the mean number of customers. It means that the mean number of customers will increase as the utilization increases.

� This research can help Abuad cafeteriatheir (Quality Of Service), by anticipating if there are many customers in the queue

� By anticipating the huge number of customers coming and going in a day, the restaurant can set a target profit that should be achieved daily.

ACKNOWLEDGEMENTS � Afe Babalola University Ado Ekiti, Ekiti State

(ABUAD) cafeteria � Operations Research Students of Department of

Accounting, Economics and Business Administration 2016/2017 session

REFERENCES 1. K. Kharwat, “Computer Simulation: an Important

Tool in the Fast-Food Industry,” Proceedings of the 1991 Winter Simulation Conference, IEEE Press, Dec. 1991, pp. 811-815.

2. D. M. Brann and B. C. Kulick, “Simulation of restaurant operations using the Restaurant Modeling Studio,” Proceedings of the 2002 Winter Simulation Conference, IEEE Press, Dec. 2002, pp. 14481453.

3. Dharmawirya, M., & Adi, E., “Crestaurant queuing model” 2011 international



cafeteria is very large ). This is only utilization rate during lunch

. proportional with the

mean number of customers. It means that the mean number of customers will increase as the

Abuad cafeteria to increase their (Quality Of Service), by anticipating if there

By anticipating the huge number of customers coming and going in a day, the restaurant can set a target profit that should be achieved daily.

Afe Babalola University Ado Ekiti, Ekiti State

search Students of Department of Accounting, Economics and Business

K. Kharwat, “Computer Simulation: an Important Food Industry,” Proceedings of

the 1991 Winter Simulation Conference, IEEE

C. Kulick, “Simulation of restaurant operations using the Restaurant Modeling Studio,” Proceedings of the 2002 Winter Simulation Conference, IEEE Press, Dec.

Adi, E., “Case study for restaurant queuing model” 2011 international

conference on management and artificial intelligence. IPEDR vol.6 (2011), pp. 52

4. Haghighi, M., Dorosti, A.,Rahnama, A., &Hoseinpour, A. “Evaluation of factors affecting customer loyalty in restaurant industry”. African Journal of Business Management, vol. 6(14), April, 2012, pp. 5041.

5. Little, J. D. C., “A proof for the queuing Operations Research, vol. 9(3), 1961, pp. 383doi: 10.2307/167570.

6. Mishra, G. D., Chauhan, V.study of service in restaurant by using queuing model”. The Bulletin of society for mathematical service and standards vol. 5 (2013), pp. 14

7. Norhati, I & Nur Hafisah,setting for learning on campus usage and performance procedia-social and behavioral science vol105,no. 2013,pp,344

8. Raman, S & Chinniah,investigation on higher learning students satisfaction on food service at university cafeteria’’ proceeding paper at the 5conference of Asian AcBusiness Cambodia.

9. S. A. Curin, J. S. Vosko, E. W. Chan, and O. Tsimhoni, “Reducing Service Time at a Busy Fast Food Restaurant on Campus,” Proceedings of the 2005 Winter Simulation Conference, IEEE Press, Dec. 2005


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Application of One Server Queuing Models to Customers Management in the Cafeteria A Case Study of...

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